The Ponzano-Regge model [1]} is a realization of (REF ) based on the triangulation of the manifold, where \(\rho \)  is given by the \(\mathrm {SU}(2)\)  irreducible representation labelled by spin \(j\in {\mathbb {N}}/2\) . \(j\)  is interpreted into the edge length suggested by the LQG framework. The vertex amplitude of the Ponzano-Regge state-sum geometrically describes the 3D geometry of the tetrahedron it is associated to, and an edge amplitude describes how two neighbouring tetrahedra are glued together.
